The Japanese Theorem for Nonconvex Polygons - A Japanese Temple Problem

A Japanese Temple Problem

In 1800, Ryōkwan Maruyama inscribed a geometry problem onto a wooden tablet and hung it in the Tsuruoka-Sannōsha shrine in what was then the Uzen Province in the northern part of Japan's Honshu island ([AUM3, Hay]). While posting a mathematical puzzler in a Shinto shrine sounds strange to us today, it was not uncommon at that time in Japan.

Japan was completely closed off from the outside world for almost the entire Edo period (1603–1868) in which Japan was ruled by the Tokugawa shogunate. Almost nothing penetrated the enforced isolation of the country, not even mathematical ideas such as the calculus of Newton, Leibniz, and Euler.

However, during this time the Japanese continued to study mathematics, or wasan, as it was called. There developed a curious tradition of inscribing geometry problems, complete with elegant and colorful figures, on wooden boards and posting them in Shinto shrines and Buddhist temples. They were gifts to the gods as well as mathematical challenges to the visitors. The boards were known as sangaku, which means "mathematical tablet." Approximately 900 sangaku survive today, but it is believed that thousands more have been lost. For more information about sangaku see [FP] or [FR].

Maruyama's sangaku is one of those that disappeared, but fortunately the inscription was recorded in the second volume of Kagen Fujita's 1807 Zoku-Sinpeki-Sanpō ([AUM3, Fu]). As was tradition, Maruyama also included the answer to the problem and a brief description of how to arrive at this answer, called the art of the problem. But he included no proof. According to [Fu] the inscription read:

Problem: draw six lines in the circle and make four circles inscribed in three of the lines. If the diameter of the southern, eastern, and western circle is 1 sun [a traditional Japanese unit of measure which is approximately 1.19 inches], 2 suns, and 3 suns, respectively, how long is the diameter of the northern circle?

Answer: 4 suns.

Art: Add the diameter of the western circle to that of the eastern one and subtract that of the southern one from it, and you will get that of the northern one. End.

Figure 1

According to [FP], the first known proof of this sangaku problem is found in the undated manuscript [Yo].

David Richeson, "The Japanese Theorem for Nonconvex Polygons - A Japanese Temple Problem," Loci (December 2013)